Need help converting this to Polar integral and evaluating it

Question

I have to convert this to polar integral and evaluate it.

$$\int _{-1}^0\int _{-\sqrt{1-x^2}}^0\:\frac{2}{1\:+\:\sqrt{x^2\:+\:y^2}}\:dy\:dx$$

I attempted the conversion and ended up with this

$$\int _{\pi }^{\frac{3\pi }{2}}\int _0^1\:\:2r\:\frac{1}{1\:+\:r}\:dr\:d\theta $$

Now, I'm stuck. Integration by parts does seem to be cooperating on this.

Answer 1

Your conversion to polar coordinates is correct. The angular integration is trivial and is simply $\pi/2$. Thus, the result is $$\begin{align}\pi \int_0^1\left (\frac{r}{1+r}\right)\,dr&=\pi \int_0^1 \left(1-\frac{1}{1+r}\right)\,dr\\\\&=\pi(1-\ln(2))\end{align}$$